Quasinormal extensions of subnormal operator-weighted composition operators in $\ell^2$-spaces

球面拟正规元组：$n$次根问题与遗传性质

Spherically quasinormal tuples: $n$-th root problem and hereditary properties

该数学研究给出球面拟正规元组的多种刻画，证明其幂次、次正规n次根均保持球面拟正规性，解决了相关多变量问题，并探究了该类元组与其极小正规扩张、对偶的联系及相关性质。

The Cauchy dual and 2-isometric liftings of concave operators

We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied.

On normal, formally normal and quasinormal composition operators in $\ell^2$-spaces

Unbounded composition operators in $L^2$-space over discrete measure spaces are investigated. Normal, formally normal and quasinormal composition operators acting in $L^2$-spaces of this kind are characterized.

On Unbounded Composition Operators in $L^2$-Spaces

Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied.

Subnormality for arbitrary powers of 2-variable weighted shifts whose restrictions to a large invariant subspace are tensor products

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invarian…

On algebras associated with invariant means on the subnormal subgroups of an amenable group

Let $G$ be an amenable group. We define and study an algebra $\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely…

单侧加权移位的多项式嵌入到2变量加权移位中

Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts

研究通过多项式嵌入将单侧加权移位表示为2变量加权移位，并探讨其子正规性及伯杰测度计算方法，回答了三个关键问题。

Joint backward extension property for weighted shifts on directed trees

Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is sat…

When is hyponormality for 2-variable weighted shifts invariant under powers?

For 2-variable weighted shifts W_{(α,β)}(T_1, T_2) we study the invariance of (joint) k- hyponormality under the action (h,\ell) -> W_{(α,β)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We show that for every k >= 1 there exists W_{(α,β)}(T_1, T_2) such that W_{(α,β)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but W_{(α,β)}(T_1, T_2) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invar…

Backward extensions of weighted shifts on directed trees

The weighted shifts are long known and important class of operators. One of known generalisation of this class are weighted shifts on directed trees, where we replace the linear order of coordinates in $\ell^2$ with a possibly more sophisticated graph structure. In this paper we focus on the question whether a weighted shift on a directed tree admits a subnormal or just power hyponormal (i.e. all powers of the operator are hyponormal) backward extension (a shift on larger directed tree). It comes out that in both cases the question whether we can obtain "joint extension" for a family of trees …

交换算子对的半亚正则性

Semi-hyponormality of commuting pairs of Hilbert space operators

本文通过构造正2×2算子矩阵的平方根公式，研究了交换算子对的半亚正则性，分析了3参数家族W_(α,β)(a,x,y)在单位立方体内的亚正则性、亚正则性、半亚正则性和弱亚正则性区域，并指出弱亚正则性与半亚正则性不重合的子区域。